Q-analogue of the Krawtchouk and Meixner orthogonal polynomials

Abstract

The comparative analysis of Krawtchouk polynomials on a uniform grid with Wigner D-functions for the SU(2) group is done. As a result the partnership between corresponding properties of the polynomials and D-functions is established giving the group-theoretical interpretation of the properties of the Krawtchouk polynomials. In order to extend such an analysis to the quantum groups SU q (2) and SU q (1, 1), q-analogues of Krawtchouk and Meixner polynomials of a discrete variable are studied in detail as solutions of the finite-difference equation of hypergeometric type on the nonuniform grid x( s) = q 2 s . However, on this grid there are two kinds of Krawtchouk and Meixner polynomials, characterized by different weight functions in the orthogonality relation. As a first step to such analysis the simpler polynomials of the first kind are considered. The total set of characteristics of these polynomials (orthogonality condition, normalization factor, recurrent relation, the explicit analytic expression, the Rodrigues formula, the difference derivative formula, various particular cases and values) is calculated.